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Zap. Nauchn. Sem. POMI, 2009, Volume 371, Pages 18–36 (Mi znsl3542)  

On approximating periodic functions by Riesz sums

N. Yu. Dodonov, V. V. Zhuk

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: Let $C$ be the space of continuous $2\pi$-periodic functions $f$ with the norm $\|f\|=\max_{x\in\mathbb R}|f(x)|$, $\sigma_n(f)$ be the Fejér sums of $f$, $E_n(f)$ be the best approximation, and let
\begin{align*} &X_n(f,a,x)=f(x)-\sigma_n(f,x)
&+\frac1{\pi(n+1)}\int^\infty_{a/(n+1)}\frac{f(x+t)+f(x-t)-2f(x)}{t^2} dt
&+\frac2\pi\sum^\infty_{k=1}\frac{(-1)^ka^{2k-1}\{(E-\sigma_n)^{2k}(f,x)\}}{(2k)!(2k-1)}. \end{align*}
Generalizing earlier results by M. Zamanskii and A. V. Efimov, the second author proved that for $f\in C$, the following relation is valid:
\begin{equation} \|X_n(f,a)\|\le C(a)E_n(f). \end{equation}
The present paper establishes advanced analogs of inequality (1) for the Riesz sums. Bibl. – 9 titles.

Key words and phrases: periodic function, $L_p$ space, Fejér sums, Riesz sums, asymptotic formulas, best approximation.

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English version:
Journal of Mathematical Sciences (New York), 2010, 166:2, 134–144

UDC: 517.5
Received: 10.11.2009

Citation: N. Yu. Dodonov, V. V. Zhuk, “On approximating periodic functions by Riesz sums”, Analytical theory of numbers and theory of functions. Part 24, Zap. Nauchn. Sem. POMI, 371, POMI, St. Petersburg, 2009, 18–36; J. Math. Sci. (N. Y.), 166:2 (2010), 134–144

Citation in format AMSBIB
\Bibitem{DodZhu09}
\by N.~Yu.~Dodonov, V.~V.~Zhuk
\paper On approximating periodic functions by Riesz sums
\inbook Analytical theory of numbers and theory of functions. Part~24
\serial Zap. Nauchn. Sem. POMI
\yr 2009
\vol 371
\pages 18--36
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl3542}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2010
\vol 166
\issue 2
\pages 134--144
\crossref{https://doi.org/10.1007/s10958-010-9853-9}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77952031512}


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